9 -1 DECIM ALS —Their PLACE in the W ORLD
M ath 2 1 0
Definition: DECIM AL NUM BERS
If x is any w hole num ber,
x . d 1 d 2 d 3 d 4 ...
represent s t he num ber:
x
+ d 1 /1 0
( w hic h is also know n as:
x
+ d 1 x 1 0 !1
so, e.g.: 3 2 1 .5 0 6 7 8 =
321
+ 5 /1 0
f ully ex panded: 3 x 10 2 + 2 x 10 1 + 1 x 10 0 + 5 x 1 0 – 1
+ d 2 /1 0 0
+ d 2 x 1 0 !2
+ 0 /1 0 0
+ 0 x1 0 – 2
+
+
+
+
d 3 /1 0 0 0
d 3 x 1 0 !3
6 /1 0 0 0
6 x1 0 – 3
+
+
+
+
F8
d 4 /1 0 0 0 0
d 4 x 1 0 !4
7 /1 0 0 0 0
7 x1 0 – 4
+
+
+
+
d 5 / 1 0 0 0 0 0 . ..
d 5 x 1 0 !5 . ..
)
8 /1 0 0 0 0 0
8 x1 0 – 5
The " ." (in 3 2 1 .5 0 6 7 8 , e.g.) is called t he decimal point.
The place v alues of t he digit s t o t he right of t he decim al point are t ent hs, hundredt hs, t housandt hs, t ent housandt hs, hundred-t housandt hs, m illiont hs, et c.
(Not to be confused w ith tens, hundreds, thousands, etc.)
In general: the place value of t he digit in t he nth decimal place (n places after the decimal point) is 1 0 – n.
Not ice t his is not sy m m et ric ... t he f irst place t o t he lef t of t he decim al point is 1 0 0 , not 1 0 1 ...!
Read and ex pand int o long form (w ords): 7 8 6 .3 0 0 7
OPERATIONS ON DECIM AL NUM BERS
W e k now how t o add, subt rac t , m ult iply and div ide decim als; w e now look at w hy .
A dd: 2 0 3 .6 0 2 + 3 .4 7
2 0 3 .6 0 2
+
=
2 0 0 + 3 + 6 /1 0 + 0 /1 0 0 + 2 /1 0 0 0
+
3 .4 7
Subt rac t 2 3 .5 ! 3 .4 7
2 3 .5
– 3 .4 7
.. .W hat ' s w rong
w it h t his pict ure?
3 + 4 /1 0 + 7 /1 0 0
CHIP M ODEL M A Y HELP!
2 3 + 5 /1 0 + 0 /1 0 0
—(
3 + 4 /1 0 + 7 /1 0 0 )
2 3 .5
–
3 .4 7
M ult iplicat ion & Div ision by 1 0 ' s:
2 .1 7 x 1 0
=
=
=
( 2
2 x1 0
2 x1 0
+
+
+
2 .1 7 x 1 0 2
Sim ilarly ,
1 /1 0
(1 /1 0 )x 1 0
1
=
+
+
+
7 /1 0 0
)
(7 /1 0 0 )x 1 0
7 /1 0
x 10
( 2 + 1/10 + 7/100 ) x 100 =
=
2 1 .7
2@1 0 0 + (1 /1 0 )@1 0 0 + (7 /1 0 0 )@1 0 0
=
200 + 10 + 7
=
2 1 .7 x 1 0 ) 1 =
2 1 .7 x 1 /1 0
2 1 .7 x 1 0 – 2 =
Sim ilarly ,
M ULTIPLICA TION BY
DIVISION BY
10
10
=
=
=
=
2 1 .7 ÷ 1 0
( 20
+
1
+
7 /1 0
)
2 0 ÷1 0 + 1 ÷1 0 + (7 /1 0 ) ÷1 0
2
+
.1
+
.0 7
2 1 .7 x 1 /1 0 0 =
÷
217
10
=
2 .1 7
2 1 .7 ÷ 1 0 0 = .2 1 7
INCREA SES THE PLA CE VA LUE OF EA CH DIGIT (BY ONE PLACE).
DOES THE OPPOSITE.
M aking sense out of t he m ult iplicat ion and div ision algorit hm s:
x
1
2
4 7
5 0
2 3 .5
2 .1 7
6 4 5
3 5
0
9 9 5
Estimate t he t ot al!
(2 3 5 x 1 /1 0 )
x (2 1 7 x 1 /1 0 0 )
1 5 .2 Ž 4 6 8 0 .0 8
= (2 1 7 x 2 3 5 ) x (1 /1 0 ) x (1 /1 0 0 )
= (2 1 7 x 2 3 5 ) x (1 /1 0 0 0 )
= (2 1 7 x 2 3 5 ) / (1 0 3 )
4 6 8 0 .0 8
1 5 .2
x
10
10
4 6 8 0 0 .8
152
9 -1 A DECIM ALS AROUND the W ORLD
M ath 2 1 0
f8
Rounding decimals:
Round 4 .3 6 7 t o t he nearest hundredt h. The nearest t w o decim als in hundredt hs are 4 .3 6 and 4 .3 7 .
The nearer is 4 . 3 7 , sinc e it is .0 0 3 great er t han 4 .3 6 7 , as opposed t o 4 .3 6 , w hich is .0 0 7 less t han 4 . 3 6 7 .
This is apparent f rom a num ber line v iew :
*
*
*
*
4 .3 6 0
4 .3 6 1
4 .3 6 2
4 .3 6 3
*
4 .3 6 4
*
4 .3 6 5
*
*
*
*
*
4 .3 6 6
4 .3 6 7
4 .3 6 8
4 .3 6 9
4 .3 7 0
4 .3 6
4 .3 7
Now round 4 . 3 6 4 5 t o t he nearest hundredt h:
4 .3 6 4 5 eit her rounds dow n t o 4 .3 6 , or up t o 4 .3 7 :
4 .3 7
)))))) 4 .3 6 5 —
4 .3 6
4 .3 6 4 5
W hen a num ber is rounded, it has been replaced w it h a nearby , but less precise, v alue.
The dif f erenc e bet w een t he v alue bef ore rounding and t he rounded-of f v alue is called t he rounding error.
The problems w ith rounding: How do y ou round 4 .3 6 5 t o t he nearest hundredt h? The nearest num bers
t erm inat ing in t he hundredt hs' plac e are 4 .3 6 and 4 .3 7 , bot h .0 0 5 f rom 4 .3 6 5 . There are t hree " rounding
rules" t hat m ay be adopt ed t o cov er suc h inst ances:
1)
Round to nearest even: The m ost w idely adopt ed rule. Fav ored because about half t he t im e y ou round
up, half t he t im e y ou round dow n; so a long series of rounding t ends t o av erage out t he errors t o zero.
2)
Round to nearest odd: Sam e as abov e, but opposit e, f or c ont rary people.
3)
Round up: This is t he sim plest bec ause any decim al w it h " 5 " in t he nex t place (af t er t he one t o w hich
t he num ber is being rounded) get s rounded up. The disadv ant age t o t his rule is t hat by alw ay s rounding
half w ay -point s up, t ot als are t oo high on t he av erage. If y ou' re t he t ax collect or, y ou round up!
Eg. Round 1 9 5 3 9 .9 8 7 4 8 t o t he nearest :
t housandt h
t ent h
1 9 5 3 9 .9 8 7
1 9 5 4 0 .0
unit
19540.
hundred
19500
t housand
20000
Caution:
Don' t round "piecemeal":
For ex am ple, Teddy rounded 4 .9 7 4 4 8 t o 4 .9 8 t his w ay :
4 .9 7 4 4 8 . 4 .9 7 4 5 . 4 .9 7 5 . 4 .9 8
Help Teddy : If 4 .9 7 4 4 8 is t o be rounded t o t he nearest hundredt h...
4 .9 7 4 4 8 w ill eit her t runc at e t o 4 .9 7 or round up t o 4 .9 8 —w hich of t hese is closer t o 4 .9 7 4 4 8 ?
W hat is t he half w ay point bet w een 4 .9 7 and 4 .9 8 ? ...is 4 .9 7 4 4 8 abov e or below t hat " half w ay point " ?
Caution:
Don' t round too soon: ...espec ially w hen using a calculat or, don' t round unt il t he com put at ion is com plet e!
Calculat ors c arry ex t ra digit s " f or f ree" , in m ost inst ances. This m et hod m inim izes error f rom rounding. (
)
For ex am ple, t o com put e 3 .5 0 x q2 , since 3 .5 0 ex presses 3 digit s of prec ision, w e w ish t o hav e t hree
digit s in t he f inal result . If w e round t he square root of 2 t oo early , w e risk an erroneous answ er. q2 .
1 . 4 1 4 2 1 3 5 6 2 . If w e round t o 1 .4 1 , t hen m ult iply by 3 .5 0 , w e obt ain 4 .9 3 5 , w hich w e round t o 4 . 9 4 ;
how ev er, if w e carry t he ex t ra decim als and m ult iply by 3 .5 0 , w e obt ain 4 .9 4 9 7 4 6 8 , w hich rounds t o
4 . 9 5 (ev en 1 .4 1 4 x 3 .5 0 w ill giv e t his result ). Conclusion: don' t round too soon.
Caution:
Express correctly! Round 8 ,4 9 8 .9 8 5 0 6 to the nearest tenth.
8 ,5 0 0 . 0
Round 8 ,4 9 8 .9 8 5 0 6 to the nearest ten.
8 ,5 0 0
A num ber w hic h has been rounded t o 3 4 0 0 is bet w een 3 3 5 0 & 3 4 5 0 .
A num ber w hic h has been rounded t o 3 4 0 0 . or 3 4 0 0 is bet w een 3 3 9 9 .5 & 3 4 0 0 .5
9 -1 B COM PUTATIONS W ITH SCIENTIFIC NOTATION
t his m at erial w as disc ussed w eek s ago, but it m akes sense t o rev iew it now
M ATH 2 1 0
f8
540000000000000 is difficult to see. We w rite, alternately, 540,000,000,000,000 to
help count the zeroes and figure the place value. Even so, most do not fully comprehend
the w ords " billion" and " trillion" . Thus, to many of us, reading or hearing the number above
is five hundred forty trillion is not illuminating. Very large numbers (as w ell as very small
numbers) w ritten in our traditional numeration system are cumbersome to w rite, read and
use. For purposes of expressing, multiplying and dividing such numbers, alternate means
are often employed, most commonly scientific notation—w rite as x.ddd... x 10 p .
(w here x may not be 0.)
Use of scientific notation requires:
• recognition of place value expressed in exponential form,
• ability to round decimal numbers,
• full command of the arithmetic properties of multiplication and division,
• understanding of significant digits (this relates to rounding).
Write in scientific notation form. (Check by multiplying!)
Key idea: Focus on the
leading digit . If it is in
the right place, the rest
w ill follow !
7,000,000 = 7 x 10 6
7,650,000 = 7.65 x 10 6
.000007 = 7 x 10 – 6
.0129 = 1.29 x 10 – 2
Examples:
Try it:
¹ Fill in the
540,000,000,000 = 5.4 x 10
¹ exponents
.00000713 = 7.13 x 10
10.02 x 10 4 =
.0001 =
... And be able to convert from Scientific Notation to Standard Form:
3.784 x 10 3 =
3784
3.702 x 10 – 3 =
4.50 x 10 – 2 =
0 .0 0 3 7 0 2
4.50 x 10 3 =
0 .0 4 5 0
4500
Arithmetic in Scientific Notation:
(3.12
x
* (4.34
10 7 )x(4.25 x 10 – 4 )
x
10 3 )3 / (2.728
x
=
(3.12
10 – 2 )
x
4.25) x (10 7
x
10 – 4 ) Ñ
(1.326 x 10 )
4
= (4.34 3 /2.728) x (10 9 /10
= 29.965727þ
x
10
11
– 2
)
Ñ
30.0 x 10 1 2
¸ No knees!:
* 5.67
x
10 1 4 +
3.33
x
10 1 5 Ñ
3.897 x 10 1 5 Ñ 3.90 x 10 1 5
¸ Use your calculator to assist in computing:
¸ *
(5.407
x
10 7 )2 x (3.66
x
10 – 4 ) ÷ (1.3311
x
10 2 )3 =
45.369268x10 4
¸ Express the result w ith correct number of significant digits in scientific notation.
( 45.369268x10 4 Ñ 4.54x10 5 )
9 -2 Rationals & Decimals
Not e t here are t hree f orm s of decim als:
Ú
RA T ION A L
M ath 2 1 0
t erm inat ing (.0 2 5 )
non-t erm inat ing repeat ing (.3 3 3 3 þ) and
non-t erm inat ing non-repeat ing (.4 1 7 4 1 1 7 4 1 1 1 7 4 1 1 1 1 7 þ)
f8
NU M BERS M A Y BE EX PRESSED A S D ECIM A LS
Converting rational fraction to decimal form: Use t he long div ision algorit hm . E.g.
7 =
1 =
3 =
3 =
1 =
9 =
3 =
10
2
4
25
125
16
64
2 =
5 =
4 =
2 =
5 =
1 =
1 =
3
9
7
11
13
75
19
A rat ional f rac t ion in reduced f orm is equiv alent t o a t erm inat ing decim al if , and only if , t he denom inat or (of t he
reduc ed f rac t ion) has only 2 ' s and/or 5 ' s as prim e f act ors. For ex am ple, 1 /1 2 5 t erm inat es in t hree places
because
1
1
1 @2 @2 @2
8
... w hich happens because 1 0 is a m ult iple of 5
)))
=
)))) = ))))))))) =
)))) =
.0 0 8
125
5 @5 @5
5 @5 @5 @2 @2 @2
1000
(.. . and, t hus, 1 0 0 0 is a m ult iple of 5 3 )
If a rat ional f rac t ion in reduc ed f orm has denom inat or w it h prim e f act ors ot her t han 2 and 5 , t he decim al does
For example: Can
not t erm inat e.
1
3
be w ritten as x ?
10
... as x ? ...
x ?
100
1000
Rat ionals w rit t en as decim als m ay be T ERM IN A T IN G (e.g. 3 /5 = .6 ) or N ON - T ERM IN A T IN G (e.g. 3 /1 1 = .2 7 2 7 2 @@@ ).
Rat ionals w hic h, reduced, hav e denom inat ors w it h prim e f act ors 2 & 5 only are T ERM IN A T IN G decim als because:
Ot her rat ionals hav e
Ú A D ECIM A L
’
THE
N ON - T ERM IN A T IN G , REPEA T IN G
decim al ex pansions—because:
N UM ERA L M A Y BE EX PRESSED A S RA T IO OF T W O IN T EGERS ,
D ECIM A L
IF...
IS T ERM IN A T IN G :
A t erm inat ing dec im al has a f init e num ber of digit s t o t he right of t he decim al point .
Such dec im als are easily ex pressed in rat ional f orm (rat io of t w o int egers); just read t hem aloud.
E.g.: 3 .4 7 =
1 .6
3 + 4 /1 0 + 7 /1 0 0
=
3 0 0 /1 0 0 + 4 0 /1 0 0 + 7 /1 0 0
=
(3 0 0 + 4 0 + 7 )/1 0 0
=
3 4 7 /1 0 0
=
You can m ak e ev en short er w ork of suc h c onv ersions: 5 2 .0 3 0 2 = 5 2 0 3 0 2 /
If a dec im al num ber t erm inat es in t he nt h place af t er t he decim al, it is equiv alent t o a f ract ion w hose num erat or
is t he num ber f orm ed by t he digit s w it hout t he decim al point , and denom inat or 1 0 n (w here n is num ber of digit s
af t er t he decim al point ) .
’ T H E D ECIM A L IS NON - T ERM IN A T IN G REPEA T IN G :
Call t he repeat ing (" unk now n" ) num ber x .
M ult iply x by 1 0 n w here n is t he lengt h of t he repet end (t he num ber of digit s t hat repeat ).
Subt rac t x f rom 1 0 nx t o obt ain (1 0 n-1 )x .
Solv e f or x ; m ult iply result ing f rac t ion by 1 0 p/ 1 0 p if necessary t o clear a decim al.
For ex am ple: x = 5 2 .1 9 0 9 0 9 0 9 0 9 0 9 0 ...
100x
– x
99x
=
=
=
5 2 1 9 .0 9 0 9 0 9 0 9 0 ...
5 2 .1 9 0 9 0 9 0 9 0 .. .
so x =
5 1 6 6 .9
99
=
=
By t he w ay , not ic e t he ex pression of a rat ional num ber as a decim al is not unique. .9
& =
Ev ery t erm inat ing dec im al has an alt ernat e f orm . EG 5 .6 8 7 = 5 .6 8 6 9 9 9 9 9 9 9 9 9 þ
T H E SET OF
TERM INATING
& REPEATING DECIM ALS IS EQ U IV A LEN T T O T H E SET OF RATIONAL N U M BERS
9-3 REAL NUMBERS!!!
8
The Pythagoreans1 believed that " Number rules the universe" ;
everything can be described in terms of numbers, and, further, that
b
c
all numbers are rational– integers, or ratios of integers.
Pythagoras proved2 w hat is called the Pythagorean Theorem3 :
a
a2 = b2 + c2 the sum of the squares of the sides of a right triangle is the square of the hypotenuse.
(and this happens only in a right triangle.)
Pythagoras' view of the describability of the universe in terms of rational numbers w as contradicted!
dest royed!by the very theorem w hich now bears his name. For if w e construct a right triangle w ith sides
of equal length 1, then w e can demonstrate that the hypotenuse (w hose length ought to correspond to
some number), is %&
2 , and %&
2 cannot be expressed as a rational number—cannot be expressed as the
ratio of tw o integers. As w e argue below :
First w e not e t hat ev ery w hole num ber has a unique prim e f ac t orizat ion.
T hen w 2 m ust hav e a prim e f ac t orizat ion w it h t he sam e prim es t o ev en pow ers:
r r r r
r
p 11 p 2 2 p 3 3p 4 4 @@@ p k k
2r 2r 2r 2r
2r
w 2 = p 1 1 p 2 2 p 3 3p 4 4@@@ p k k
w =
%&
2 = p/q
Suppose
w here p,q , Z, q … 0 [af t er all, t hat ' s w hat it w ould m ean f or % &
2 t o be rat ional] .
... and w e m ay assum e p/q has been reduc ed t o low est t erm s (so p and q hav e NO c om m on f ac t ors).
M ult iply ing bot h sides by q and squaring giv es us
2 q2 = p2 .
W hic h m eans 2 m ust be a f ac t or of p 2 , w hic h in t urn im plies 2 m ust be a f ac t or of p.
So p 2 has an odd pow er of 2 in it s prim e f ac t orizat ion. .. This is not possible.
Sinc e ev ery part of our argum ent is t rue, t he only f law m ust be in t he assum pt ion w e m ade in t he f irst place, t hat % &
2 can
be w rit t en in rat ional f orm . ]
Pythagoras and his follow ers w ere devastated. Even today, w e are troubled to discover there are
numbers
w hich cannot be w ritten in the friendly form of a ratio of integers—but the irritation is diminished by the
comf orting thought that although such numbers exist, they are relatively few , and may be mostly
ignored. NOT!
Let' s make a small digression. We now know these irrationals exist, and %&
2 is one of them, and can
not be w ritten as a ratio of integers; but w hat is the nature of an irrational? What is the decimal
representation of an irrational number? We have previously seen that a rational number can be
expressed in decimal form using the division algorithm, and that the decimal form either terminates or
repeats.
Furthermore, any terminating or repeating decimal can be expressed as a ratio of tw o integers.
Therefore: The set of rational numbers is exactly the set of terminating and repeating decimals.
The Irrationals (reals not expressible as ratio of tw o integers) are non-terminating, non-repeating
decimals.
Consider, e.g.
... or
®
1.10200300040000500000600000070000000800000000900000000010000000000011...
1.12123123412345123456123456712345678123456789123456789101234567891011...
The sum of tw o rationals is rational. What about the sum of tw o irrationals?
This can be a little difficult to explore, since w e have no general arithmetic algorithms for irrationals.
What is %&
2 + %&
3 ? (Hint: square that!)
Is the sum of tw o irrationals alw ays irrational?
...How about %&
2 + %&
8 ?
!3 %&
2 + %&
1&
8?
What is the sum of a rational number and an irrational? Consider, for example, %&
2 + ½.
Intuitively, w e have a non-terminating non-repeating decimal plus .5, the total of w hich should be
non-terminating non-repeating. How ever, w e look for proof....
1
2
3
st uden t s/ disc iples o f t he sc hool of Py t hag oras at Cro t ona in So ut hern It aly
. . . or so he is c redit ed. Py t hagoras m ay hav e prov ed it , perhaps one or som e of his st udent f ollow ers.
t hough it w as k now n t o t he Baby lonians & Egy pt ians at least c ent uries earlier
We can easily prove %&
2 + ½ is irrational:
Suppose %&
2 + ½ = a/b ... w here a/b is rational.
Then %&
2 = a/b – ½ w hich, since it is the difference of tw o rational numbers, is a rational number.
This contradicts the know n status of %&
2 – it’ s irrational.
The argument above show ing %&
2 is irrational may be used to show that the square root of any prime (or
of any composite number w hose prime factors are not all to an even pow er) is not rational.
Furthermore, there are other irrational numbers (pi and e for instance) that " occur" in numerous
circumstances. In addition, for every irrational, w e can construct a w hole family of irrationals by adding
any rational. (Irrational + rational is ...) Thus w e can easily see there are at least as many irrational
numbers as rationals. It w as not until the end of the nineteenth century that a means of tackling the
cardinality of infinite sets w as devised to settle this question: How does the size of the set of irrationals
compare with the size of the set of rationals?
Georg Cantor4 w as primarily responsible for the development of theory of sets & classes, w ith particular
emphasis on the infinite. He devised the means of comparing cardinality of sets by mapping, or
establishing a 1-1 correspondence betw een tw o sets to show they are of the same cardinality. (Recall
our demonstrations that the cardinality of N is equivalent to that of Z.) Although the rationals are dense
in the number line, the cardinality of Q is the same as that of N! But the cardinality of the irrationals is
greater! Here is Cantor' s ingeniously simple proof:
Any set w hose cardinality is that of N can be listed. (The act of listing establishes a 1-1
correspondence w ith N.) Suppose a " list" of all irrationals is presented. We w ill show that the
list does not—CANNOT—contain all the irrationals, by constructing an irrational number that is not
in the list.
Suppose t he f irst f ew num bers
in t he alleged list are:
m.d1 d2 d3 d4 d5 ...
n.e1 e2 e3 e4 e5 ...
o.f 1 f 2 f 3 f 4 f 5 ...
p.g1 g2 g3 g4 g5 ...
e.g.
10.1450368...
4.2907863...
.0006721...
.0332737...
.3
7
2
4
"
We select a digit different from d1 as the first decimal digit of our number; select a digit different
from e2 as the second digit of our number; select a digit different from f 3 as the third digit, and so
on. The number so constructed cannot be the first number in the list because it differs in the first
decimal place; cannot be the second number in the list because it differs in the second decimal
place from that number; and so on. So it is not in the list at all! Thus the list is not complete.
Therefore, it is not possible to list all the irrationals, and thus they cannot be put into 1-1
correspondence w ith the natural numbers. The cardinality of the set of irrationals is greater than
the cardinality of N. Thus although the rationals are dense in the number line (no interval gaps),
there are other numbers (irrationals) in the number line, and they far outnumber the rationals!
Properties of Arithmetic Operations on REAL numbers:
Addition and multiplication on the set of all real numbers have the follow ing properties:
CLOSURE; COM M UTA TIV ITY ; A SSOCIA TIV ITY ; IDENTITY (0 for + ; 1 for x );
INV ERSES for + : for any decimal a, -a is the additive inverse.
INV ERSES for x :
for each decimal b other than 0, there is another decimal number, 1/b such that bx (1/b) = 1
Subtraction and Division have the closure property except for division by 0.
4
1 8 4 5 -1 9 1 8 Can t o r' s t h eo ry o f t h e in f in it e so ro c k ed t h e sc ien t if ic an d , part ic u larly , m at h em at ic al w o rld o f h is t im e t h at so m e
ant agonist s w ere able t o bloc k his adv anc em ent ; he w as v iew ed as subv ersiv e by som e, unbalanc ed by ot hers. T he bit t erness of his lif e,
c oupled w it h insec urit y bred in Cant or, and perhaps som e genet ic predisposit ion, led t o a num ber of break dow ns f or w hic h Cant or w as
hospit alized. In t he early t w ent iet h c ent ury his w ork began t o be rec ognized as t he prof oundly real w ork of a genius; t his rec ognit ion w as
t oo lit t le and t oo lat e.
. Our new Universe:
N W Z Q R
.
N
.
W =
{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, @@@ }
.
Z
{ @@@ , !5, !4, !3, !2, !1, 0, 1, 2, 3, 4, 5, @@@ }
.
Q =
=
=
.
R
=
=
=
=
=
=
=
{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, @@@ }
{ p/ q | p,Z, q,Z, and q … 0 }
{ 0, 1 / 1 , !1 / 1 , !1 / 2 , 1 / 2 , 1 / 3 , !1 / 3 , 2 / 3 , 2 / 1 , !2 / 1 , !2 / 3 , @@@ }
The set of all terminating and repeating decimals
N
W
Z
Q
The set of all decimals
R
The set containing all rationals and all irrationals
Q cI
Set of all decimals that terminate or repeat c Set of all nonterminating nonrepeating decimals
Set of all decimals that can
c
Set of all decimals that cannot
9 be w ritten as a ratio of integers A
9
be w ritten as a ratio of integers A
Note:
If w e think of R as our new Universe, I, the set of irrationals, is the complement of Q, the set of rationals.
Every real number is rational or irrational– one or the other/
13-3
Some details & footnotes
Math 210
.2
Trichotomy:
Given any tw o real numbers a and b, exactly one of the follow ing holds: a < b or a > b or a = b.
In particular, given any real number r, then either r is positive (r > 0), or r is negative (r < 0), or r is 0.
Some important details:
"
The irrationals form a set disjoint (separate) from the rationals.
"
Together the rationals and irrationals make up R, the set of real numbers.
"
N f W f Z f Q f R. (Each of these is contained in —is part of —the next.)
"
The set of real numbers is larger than the set of rationals, and is dense in the number line!
"
The set of real numbers can be thought of as corresponding to the points (all points) on a line.
(The set of real numbers may also be thought of as the set of all decimal numbers, including those
w hich terminate or repeat (rationals) and those w hich neither terminate nor repeat ( irrationals). )
Radicals & Fractional exponents:
%, called radical, stands for the principle square root—the positive root of a positive number.
To indicate the negative of the square root, w e w rite !% . For instance, %16 = 4. !%16 = -4.
1 /2
%&
x may be w ritten in the form x .
Notice that
%&
x %&
x =
x (provided %&
x exists)... And this can also be w ritten
x 1 /2 x 1 /2 = x
(STRICTLY Optional !!) More Radicals & Fractional exponents:
The %, or radical, symbol is used to denote other roots of numbers, such as cube root:
3
%&
2&
7 =
3
We define: n% denotes the nth root of a number. For instance, 6 %64 = 2 because 2 6 = 64.
3
1 /3
Just as %x may be w ritten in the form x 1 /2 ,
%&
x is x
and in general: x 1 /q = q %&
x.
%&
8 = 2
x p/q = (x 1 /q)p = (q%x)p .
32 2 /5 = (32 1 /5 )2 = (2)2 = 4
= 1/ %8 = 1/2
32 !2 /5 = 1/(32 1 /5 )2 = 1/(2)2 = 1/4
...and more generally:
8 1 /3 =
E.g.
8
!1 /3
3
3
QUIZ YOURSELF! (1-15 simplify; 16-20 answ er rational, irrational, unpredictable)(* = optional !)
1*. 16 1 /4 =
2*. 32 1 /5 =
3*. 27 1 /3 =
4*. 64 1 /3 =
5. 4 !1 /2 =
6. 36@36 !1 /2 =
7. 7 1 /2 @7 1 /2 =
8*. 4 3 /2 =
9*. (a/ b )!1 /2 =
10*. (16/9)!1 /2 =
11. %x 6 =
12. %72a6 b7 =
13. %72 + %18 =
14. %36@2 3 @x 1 2 y 3 =
15. { 64x 5 y 6 @(x+ y)8 } 1 /2 =
16. The sum of a rational and an irrational is:
17. The product of a rational and irrational is:
18. The sum (difference) of tw o rationals is:
19. The sum (difference) of tw o irrationals is:
20. The product (quot ient ) of tw o rationals is:
21. The product (quot ient ) of tw o irrationals is:
¿ Answ ers?:
1. 2 2. 2 3. 3
4. 4
5. 1/2
6. 6
7. 7
8. 8
9. (b / a)1 /2 or
12. %8@9a6 b7 = %4@9a6 b6 2b = 2@3a3 b3 %2b 13. %9@4@2 + %9@2
14. %2 2 @3 2 @2 3 @x 1 2 y 3 = %2 5 @3 2 @(x 6 )2 @y 3 = 4@3x 6 y%2y = 12x 6 y%2y
=
%b
/ %a 10. 3/4
6%2 + 3%2
=
11.
%(x 3 )2 = x 3
9%2
15. { 64x 5 y 6 @(x+ y)6 } 1 /2 = { 2 6 x 5 y 6 @(x+ y)6 } 1 /2 = 8x 2 y 3 (x+ y)4 { x} 1 /2 16. irrational (see prior discussion)
17. if the rational is zero, the product is rational because it' s 0; otherw ise, irrational.
(Consider, eg, 2@%2, and.... If q , Q and q@b = p , Q then b = p/q must be rational unless q is zero.)
18. rational (+ or !) 19. unpredictable (+ or !) (Consider %2 + %2 = 2%2; (9 ! %2) + (%2 ! 5)
20. rational (@ or ÷) 21. unpredictable (@ or ÷) (Consider %2@%2 = 2; %2@%3 = %6)
=
4)